On realizations of representations of the infinite symmetric group

Denote by $\mathbb N$ the set of positive integers $\{1,2,\dots\}$. Let $\mathfrak S_\mathbb X$ stand for the group of all finite permutations of the set $\mathbb X=-\mathbb N\cup\mathbb N$. Consider the subgroups
$$\mathfrak S_\mathbb N=\{s\in\mathfrak S_\mathbb X\colon s(-k)=-k\text{ for all }k\in\mathbb N\}$$
and
$$\mathfrak D=\{s\in\mathfrak S_\mathbb X\colon -s(k)=s(-k)\text{ and }s(\mathbb N)=\mathbb N\}.$$
Given a spherical representation $\pi$ of the pair $(\mathfrak S_\mathbb N\cdot\mathfrak S_{-\mathbb N},\mathfrak D)$, we construct a spherical representation $\Pi$ of the pair $(\mathfrak S_\mathbb X,\mathfrak D)$ such that the restriction of $\Pi$ to the group $\mathfrak S_\mathbb N\cdot\mathfrak S_{-\mathbb N}$ coincides with $\pi$.